We’ve seen how to use the method of undetermined coefficients and the method of variation of parameters to compute the general solution to a nonhomogeneous system of differential equations. We can also use the matrix exponential, e^(At), where A is an n x n matrix of constants, as part of the following formula for the solution to a nonhomogeneous system.
Read MoreWe’ll learn much more about matrices in Linear Algebra. For now, we just need a brief introduction to matrices (for some, this may be a review from Precalculus), since we’ll be using them extensively to solve systems of differential equations.
Read MoreIf undetermined coefficients isn’t a viable method for solving a nonhomogeneous system of differential equations, we can always use the method of variation of parameters instead. Just like with undetermined coefficients, we have to start by finding the corresponding complementary solution, which is the general solution of the associated homogeneous equation.
Read MoreThe method of undetermined coefficients may work well when the entries of the vector F are constants, polynomials, exponentials, sines and cosines, or some combination of these. Our guesses for the particular solution will be similar to the kinds of guesses we used to solve second order nonhomogeneous equations, except that we’ll use vectors instead of constants.
Read MoreNow we want to look at the phase portraits of systems with complex Eigenvalues. The equilibrium of a system with complex Eigenvalues that have no real part is a stable center around which the trajectories revolve, without ever getting closer to or further from equilibrium. The equilibrium of a system with complex Eigenvalues with a positive real part is an unstable spiral that repels all trajectories. The equilibrium of a system with complex Eigenvalues with a negative real part is an asymptotically stable spiral that attracts all trajectories.
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